Properties

Degree 1
Conductor $ 7 \cdot 11 \cdot 13 $
Sign $-0.289 + 0.957i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + i·2-s − 3-s − 4-s + i·5-s i·6-s i·8-s + 9-s − 10-s + 12-s i·15-s + 16-s − 17-s + i·18-s i·19-s i·20-s + ⋯
L(s,χ)  = 1  + i·2-s − 3-s − 4-s + i·5-s i·6-s i·8-s + 9-s − 10-s + 12-s i·15-s + 16-s − 17-s + i·18-s i·19-s i·20-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(\chi,s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(\chi,s)\cr =\mathstrut & (-0.289 + 0.957i)\, \Lambda(\overline{\chi},1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s,\chi)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s,\chi)\cr =\mathstrut & (-0.289 + 0.957i)\, \Lambda(1-s,\overline{\chi}) \end{aligned} \]

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(1001\)    =    \(7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.289 + 0.957i$
motivic weight  =  \(0\)
character  :  $\chi_{1001} (538, \cdot )$
Sato-Tate  :  $\mu(4)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 1001,\ (1:\ ),\ -0.289 + 0.957i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.4608130408 + 0.6209949923i$
$L(\frac12,\chi)$  $\approx$  $0.4608130408 + 0.6209949923i$
$L(\chi,1)$  $\approx$  0.5126649204 + 0.3627181627i
$L(1,\chi)$  $\approx$  0.5126649204 + 0.3627181627i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−21.13243048663768171721456903720, −20.58475796010146504816363142652, −19.76593671954279410829708309946, −18.90696992482814844608066656505, −18.0401759341672142825259497453, −17.4792520382149684162253042044, −16.64287323428662124828493912121, −16.01909021261487138718118658153, −14.86231397345654279843391853961, −13.68703920321075891698333075110, −12.95100524209596681102204036189, −12.40328667356680007780517134218, −11.61675332344257478281430921227, −10.99272686843589347765051407451, −9.98951821219568718924289172741, −9.39041090655650824018588526994, −8.39566448181983589497103072120, −7.47583199561230242303149400793, −5.9565916929922609707815310610, −5.48695327609349080300305987994, −4.27574848895269687050944922654, −4.034887301403260310839638953251, −2.27579942002309684741870220354, −1.41301891087587305565588866824, −0.4204800416065974055660167811, 0.46476972629299769561378432035, 2.04942211841199842963417059221, 3.55025274284193280714776956567, 4.43807413807585319820251024769, 5.36892965162943216590627963678, 6.2015498586975722165973864703, 6.88631467503051616750986705645, 7.42050466272385359509182709334, 8.61406592604373879429728725832, 9.639052490249624484521264446251, 10.44053488773943063568832655653, 11.19563272508423226485374684101, 12.11214070535146954207454388611, 13.15544336286772719013606809860, 13.807485104977663856619118813407, 14.797212154939235214621652301426, 15.57478717536382627077042278787, 16.02013314222953958817759436660, 17.15064846411208408626388567668, 17.63999253406848531101081372018, 18.30722114500940334927058744195, 18.93796384021992827577049772721, 19.94735702930759696681614071222, 21.421207801691470620297572635982, 22.07104315002090413471643645576

Graph of the $Z$-function along the critical line